IP-sets and polynomial recurrence
We combine recurrence properties of polynomials and IP-sets and show that polynomials evaluated along IP-sequences also give rise to Poincaré sets for measure-preserving systems, that is, sets of integers along which the analogue of the Poincaré recurrence theorem holds. This is done by applying to measure-preserving transformations a limit theorem for products of appropriate powers of a commuting family of unitary operators.
Ergodic Theory and Dynamical Systems
Bergelson, V., Furstenberg, H., & McCutcheon, R. (1996). IP-sets and polynomial recurrence. Ergodic Theory and Dynamical Systems, 16 (5), 963-974. https://doi.org/10.1017/S0143385700010130